Special biserial coalgebras and representations of quantum \(SL(2)\).
Let \(K\) be a field. Following the definitions of a special biserial finite-dimensional \(K\)-algebra and a string \(K\)-algebra, the author introduces and studies the class of special biserial \(K\)-coalgebras and string \(K\)-coalgebras. Relations between these concepts and the concept of a string coalgebra introduced by the reviewer [in Lect. Notes Pure Appl. Math. 236, 465-492 (2004; Zbl 1072.16015)] are discussed, elementary properties of special biserial \(K\)-coalgebras are proved, and the theory of special biserial \(K\)-coalgebras is developed in the paper. The results are used to get an explicit description of finite-dimensional comodules and the Auslander-Reiten quiver for the coordinate Hopf algebra \(C\) of the quantum \(SL(2)\) at a root of unity. In particular, it is shown that \(C\), viewed as coalgebra, is tame of discrete comodule type. The stable Green ring of \(C\) and the quantum dimensions of indecomposable \(C\)-comodules are computed. In particular, it is shown that every finite-dimensional non-injective indecomposable \(C\)-comodule is of non-zero quantum dimension.
scientific article; zbMATH DE number 6070504
scientific article